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In this paper we develop efficient numerical schemes to solve ordinary differential equations. Our methods are of the Nordsieck type, adaptive and capable of automatically controlling the global error of a numerical solution. A special feature of the new stepsize selection algorithms introduced here is the global error estimation quality control. Two different ways of attaining the preassigned accuracy of computation are examined in the paper. Namely, we implement the global error control mechanism based on reducing the maximum stepsize bound and the other one is based on reducing the local error tolerance. An accurate starting procedure for the adaptive Nordsieck methods is presented in full detail. Our intention here is to find the most effective strategy of stepsize selection. Theoretical investigation is supplied with numerical tests. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modified inhomogeneous Helmholtz equation Δ u( x, y) u( x, y) f( x, y), x , 0 y 1, with inhomogeneous Cauchy data being given at y 0. The problem is known to be ill-posed, as the solution (if exists) does not depend continuously on the given data. We propose a regularization method to obtain a stable approximate solution of the problem and get some error estimates. Finally, a numerical example shows the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Recently, Schmitt, Weiner, and Erdmann have proposed an efficient family of numerical methods termed Implicit Parallel Peer (IPP) methods. They are a subclass of s-stage general linear methods of order s – 1. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff differential equations. The special construction of IPP methods allows for a parallel implementation, which is advantageous in modern high-performance computation environment. In this paper we add one more useful functionality to IPP methods, i.e. automatic global error control. We show that the global error estimation developed by Kulikov and Shindin in multistep formulas is suitable for the methods of Schmitt, Weiner and Erdmann. Moreover, that global error estimation can be done in parallel. An algorithm of efficient stepsize selection is also discussed here. [ABSTRACT FROM AUTHOR]

In this paper, we consider two-parameter singularly perturbed boundary value problems via fitted splines. We propose an exponentially fitting factor to the highest order derivative of the problem and discretize by parametric cubic spline functions on a uniform mesh. An error analysis of the method is presented. The efficiency of the method is demonstrated by test examples which support the theoretical results. [ABSTRACT FROM AUTHOR]

We introduce and investigate a probabilistic model of a coherent system which consists of m elements and is intended for fulfilment of homogeneous tasks. On the first stage, one task out of the total number n of tasks enters each element of the system. The result of work of an element is either a fulfilment of the task or a failure of the element. In the case of the failure of an element, it is excluded from the system, and the task which is not fulfilled returns to the queue of those waiting for fulfilment. On the second stage, m1 tasks are sent to the system, where m1 is the number of elements of the system remained operable after the first stage, and so on. In the paper, a detailed analysis of the suggested model is realised in the case where each element of the system fulfils tasks with the same probability independently of the rest of the elements. [ABSTRACT FROM AUTHOR]

The Fourier pseudo-spectral method has been studied for a one- dimensional coupled system of viscous Burgers equations. Two test problems with known exact solutions have been selected for this study. In this paper, the rate of con- vergence in time and error analysis of the solution of the first problem has been studied, while the numerical results of the second problem obtained by the present method are compared to those obtained by using the Chebyshev spectral collocation method. The numerical results show that the proposed method outperforms the conventional one in terms of accuracy and convergence rate. [ABSTRACT FROM AUTHOR]

Five different error estimation strategies suitable for the Gauss-type Nested Implicit Runge–Kutta method of order 4 have been presented and tested numerically in [Kulikov and Shindin, Lecture Notes in Computer Science: 136–143, 2007, Kulikov and Shindin, Appl. Numer. Math. 59: 707–722, 2009]. The nested Implicit Runge–Kutta schemes introduced recently are an efficient class of Implicit Runge–Kutta formulas. In this paper we deal with the methods of order 6. One scheme of such sort has been constructed in [Kulikov and Shindin, Appl. Numer. Math. 59: 707–722, 2009]. Now we present a one-parametric family of the above-mentioned formulas of order 6 by relaxing the accuracy requirement for some stage values. This allows the error estimation strategies designed for the method of order 4 to be extended to the higher-order Gauss-type Nested Implicit Runge–Kutta method. We also present the particulars of the efficient implementation of this method, which is stable and accurate. The numerical examples confirm the efficiency of the numerical scheme under consideration for both ordinary differential equations and partial differential equations. [ABSTRACT FROM AUTHOR]

This paper deals with asymptotically correct methods to evaluate the local and global errors of Nordsieck formulas applied to ordinary differential equations. It extends naturally the results developed by Kulikov and Shindin [ Comp. Math. Math. Phys. (2000) 40, 1255–1275] in local and global error computation of multistep methods, but shows that Kulikov and Shindin's technique becomes more complicated when implemented in numerical methods, for which the concepts of consistency and quasi-consistency are not equivalent (see Skeel [ SIAM J. Numer. Anal. (1976) 13, 664–685]). A new property termed super quasi-consistency is introduced and special cases of Nordsieck formulas with cheaper error estimation are found. Numerical examples are included to confirm practically the theory presented in this paper. [ABSTRACT FROM AUTHOR]

This paper investigates an error estimate proposed by Warnock and studied by Halton (2005). That error estimate is simply the sample standard error applied to certain non-randomized quasi-Monte Carlo points. This quasi-standard error (QSE) closely tracks the actual error in an example, and looks to be at least as accurate as a standard error based on random replication. We also show that the quasi-standard error is not unreasonably large in its intended use. But there are quasi-Monte Carlo (QMC) constructions for which the QSE severely underestimates the true error. Moreover, discrepancy considerations do not separate these counter-examples from other cases where the method might be reliable. We conclude that the QSE is not yet ready to be trusted in applications. [ABSTRACT FROM AUTHOR]

This paper concerns a posteriori error estimation for the primal and dual mixed finite element methods applied to the diffusion problem. The problem is considered in a general setting with inhomogeneous mixed Dirichlet-Neumann boundary conditions. New functional-type a posteriori error estimators are proposed that exhibit the ability both to indicate the local error distribution and to ensure upper bounds for discretization errors in primal and dual (flux) variables. The latter property is a direct consequence of the absence in the estimators of any mesh-dependent constants; the only constants present in the estimates stem from the Friedrichs and trace inequalities and, thus, are global and dependent solely on the domain geometry and the bounds of the diffusion matrix. The estimators are computationally cheap and require only the projections of piecewise constant functions onto the spaces of the lowest-order Raviart-Thomas or continuous piecewise linear elements. It is shown how these projections can be easily realized by simple local averaging. [ABSTRACT FROM AUTHOR]

-The main purpose of this paper is to propose the Chebyshev spectral-collocation method for a class of the weakly singular Volterra integral equations (VIEs) with proportional delay. The proposed method also are applicable to a class of the weakly singular VIEs with proportional delay possessing unsmooth solution. To provide a rigorous error analysis for the proposed method, we prove the the uniqueness and smoothness of the solution. The error analysis shows that the numerical errors decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence. [ABSTRACT FROM AUTHOR]

Numerical errors for models of the sea surface undulation are studied in the paper based on spectral decomposition of the stochastic field of water level. Such errors depend on the number of random harmonics in the spectral model and on the size of the domain for which the spectral model is constructed. Numerical errors are studied for temporal and spatial spectral models. [ABSTRACT FROM AUTHOR]

In this paper, we make an offer of the lattice approximate method for solving a class of multi-dimensional integro-differential equations with the initial conditions. Also, we analyze the worst case error measured in weighted Korobov spaces for these equations. Finally, numerical examples complete this work. [ABSTRACT FROM AUTHOR]

- This paper is concerned with the derivation of conforming and non-conforming functional a posteriori error estimates for elliptic boundary value problems in exterior domains. These estimates provide computable and guaranteed upper and lower bounds for the difference between the exact and the approximate solution of the respective problem. We extend the results from [5] to non-conforming approximations, which might not belong to the energy space and are just considered to be square integrable. Moreover, we present some numerical tests. [ABSTRACT FROM AUTHOR]

For solving linear ill-posed problems with noisy data, regularization methods are required. In this paper we study regularization under general noise assumptions containing large noise and small noise as special cases. We derive order optimal error bounds for an extended Tikhonov regularization by using some pre-smoothing. This accompanies recent results by the same authors, Regularization under general noise assumptions, Inverse Problems 27:3, 035016, 2011. [ABSTRACT FROM AUTHOR]

In this paper we use the discontinuous Galerkin finite element method for the space-semidiscretization of a nonlinear nonstationary convection--diffusion problem defined on a nonpolygonal two-dimensional domain. Using Zláámal's concept of the ideal curved elements, we define a finite element space . We prove the ''ideal'' versions of the inverse and the multiplicative trace inequalities known for standard straight triangulations. Further, we define a projection on the finite element space and study its approximation properties. The obtained results allow us to derive an H1-optimal error estimate for the discontinuous Galerkin method employing the ideal curved elements. [ABSTRACT FROM AUTHOR]

This paper develops a combined a posteriori analysis for the discretization and iteration errors in the solution of elliptic eigenvalue problems by the finite element method. The emphasis is on the iterative solution of the discretized eigenvalue problem by a Krylov-space method. The underlying theoretical framework is that of the Dual Weighted Residual (DWR) method for goal-oriented error estimation. On the basis of computable a posteriori error estimates the algebraic iteration can be adjusted to the discretization within a successive mesh adaptation process. The functionality of the proposed method is demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

A Cauchy problem for elliptic equations with nonhomogeneous Neumann data in a cylindrical domain is investigated in this paper. For the theoretical aspect the a-priori and a-posteriori parameter choice rules are suggested and the corresponding error estimates are obtained. About the numerical aspect, for a simple case results given by two methods based on the discrete Sine transform and the finite difference method are presented; an idea of left-preconditioned GMRES (Generalized Minimum Residual) method is proposed to deal with the high dimensional case to save the time; a view of dealing with a general domain is suggested. Some ill-posed problems regularized by the quasi-boundary-value method are listed and some rules of this method are suggested. [ABSTRACT FROM AUTHOR]

In this paper solutions of ill-posed problems with some a priori information about the exact solution are considered. For the first group of such problems it is supposed that the exact solution is a bounded piecewise convex function on some bounded segment [ a, b]. It is shown that the set of these functions is a compact set in LP[ a, b] and an approximate solution tends to the exact one uniformly on some subset of [ a, b]. Sourcewise represented functions form the second group of the problems. For this case it is possible to find a so-called a posteriori error estimation of an approximate solution. The method of extending compacts may help to estimate this a posteriori error. [ABSTRACT FROM AUTHOR]

We consider linear ill-posed problems in Hilbert spaces with a noisy right hand side and a given noise level. To solve non-self-adjoint problems by the (iterated) Tikhonov method, one effective rule for choosing the regularization parameter is the monotone error rule (Tautenhahn&Hämarik, Inverse Problems, 1999, 15, 1487-1505). In this paper we consider the solution of self-adjoint problems by the (iterated) Lavrentiev method and propose for parameter choice an analog of the monotone error rule. We prove under certain mild assumptions the quasi-optimality of the proposed rule guaranteeing convergence and order optimal error estimates. Numerical examples show for the proposed rule and its modifications much better performance than for the modified discrepancy principle. [ABSTRACT FROM AUTHOR]

A method of test particles is considered for the solution of a linearized Boltzmann equation using two stochastic estimates: based on time and on intersections. The main goal of the paper is the construction of upper bounds for errors of the method in the mertics of the space C of continuous functions and its optimization on the basis of the theory of Monte Carlo functional algorithms. A new universal approach for the construction of upper error bounds is used, which is applicable for any degree of dependence of solution estimates at the grid nodes. Based on these upper error bounds for two considered stochastic estimates, optimal relations (in the sense of these bounds) are constructed for the sample amount and the number of grid nodes, which guarantee that the error does not exceed a given level. The optimal relations obtained here are numerically verified on two problems: the classic problem of heat transfer between two parallel plates and the two-dimensional problem of a transversal supersonic flow of a rarefied binary gas mixture around a plate. [ABSTRACT FROM AUTHOR]

In [2] we introduced a new type of mixed finite element approximations for two- and three-dimensional problems on distorted polygonal and polyhedral meshes that consist of cells having different forms. Additional degrees of freedom that arise in the process are excluded by a special condition that is natural for the mixed finite element approximations considered. This paper is devoted to the error analysis of the respective finite element solutions. We show that under certain assumptions on the regularity of the exact solution the convergence rate for the new approximations is the same as for the Raviart-Thomas finite element approximations of the lowest order. [ABSTRACT FROM AUTHOR]

In this paper a duality-based a posteriori error analysis is developed for the conforming hp Galerkin finite element approximation of second-order elliptic problems Duality arguments combined with Galerkin orthogonalty yield representations of the error in arbitrary quantities of interest From these error estimates, criteria are derived for the simultaneous adaptation of the mesh size h and the polynomial degree p. The effectivity of this procedure is confirmed by numerical tests. [ABSTRACT FROM AUTHOR]

For a given function, we consider the problem of minimizing the P1 interpolation error on a set of triangulations with a fixed number of triangles. The minimization problem is reformulated as the problem of generating a mesh which is quasi-uniform in a specially designed metric. For functions with indefinite Hessian, we show the existence of a set of metrics with highly diverse properties. This set may include both anisotropic and isotropic metrics, which produce families of different meshes providing a comparable reduction of interpolation error. The developed theory is verified with numerical examples. [ABSTRACT FROM AUTHOR]

We study the effects caused by inexact knowledge of the Poisson's ratio in the linear elasticity problem and derive estimates for the incremental quantity that characterizes the variability of the internal energy and the logarithmic derivative of the energy with respect to the Poisson's ratio. These quantities characterize errors caused by inexact (incomplete) knowledge of material constants, which is typical for computer modelling of physical and engineering problems.We prove that their behaviour is drastically different for two different classes of boundary conditions, namely the boundary conditions of the first class generate solutions that are relatively stable with respect to small variations of the Poisson's ratio, while other conditions generate solutions, for which the quantities blow up if the ratio tends to one half. [ABSTRACT FROM AUTHOR]

Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented. [ABSTRACT FROM AUTHOR]

The problem of interaction of a thermocouple inserted into a unitary solid propellant with a moving heat wave spreading inside the propellant from the combustion surface is considered. The results of numerical modelling showed that a heat sink directed deep into the propellant occurs along the thermocouple wire because of large differences in the heat conductivity coefficients of the propellant and the thermocouple material, which essentially changes the temperature of the thermocouple junction and hence distorts the indications. [ABSTRACT FROM AUTHOR]

We consider an adaptive finite element method (AFEM) for the Laplace eigenvalue problem in bounded polygonal or polyhedral domains. We provide an a posteriori error analysis based on a residual type estimator which consists of element and face residuals. The a posteriori error analysis further involves an oscillation term. We prove a reduction in the energy norm of the discretization error and the oscillation term. Numerical results are given illustrating the performance of the AFEM. [ABSTRACT FROM AUTHOR]

We provide an a posteriori error analysis of finite element approximations of pointwise state constrained distributed optimal control problems for second order elliptic boundary value problems. In particular, we derive a residual-type a posteriori error estimator and prove its efficiency and reliability up to oscillations in the data of the problem and a consistency error term. In contrast to the case of pointwise control constraints, the analysis is more complicated, since the multipliers associated with the state constraints live in measure spaces. The analysis essentially makes use of appropriate regularizations of the multipliers both in the continuous and in the discrete regime. Numerical examples are given to illustrate the performance of the error estimator. [ABSTRACT FROM AUTHOR]

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find some unknown parameters of the model. The equation for the error of the optimal solution is derived through the statistical errors of the input data (background, observation, and model errors). A numerical algorithm is developed to construct an a posteriori covariance operator of the analysis error using the Hessian of an auxiliary optimal control problem based on the tangent linear model constraints. [ABSTRACT FROM AUTHOR]

We develop an adaptive finite element concept for elliptic optimal control problems with constraints on the state. In order to generate goal-oriented meshes, we extend the Dual Weighted Residual (DWR) concept proposed by Becker and Rannacher for PDE-constrained optimization to the state constrained case. Using the optimality system of the underlying PDE-constrained optimal control problem we obtain a representation for the error in the objectives. Based on this representation we define local error indicators, whose performance we investigate by means of a numerical example. [ABSTRACT FROM AUTHOR]

A class of multi particle Lagrangian stochastic models is considered mimicking 2 D turbulence. The maximum likelihood approach is used to estimate their parameters. An error analysis is carried out by Monte Carlo means. The method allows to estimate some physically important characteristics of Lagrangian motion such as relative dispersion and Lyapunov exponent by observing only one particle pair. An illustrative example is given based on real data. [ABSTRACT FROM AUTHOR]

The subject-matter is the analysis of the discontinuous Galerkin finite element method of lines applied to a linear nonstationary convection-diffusion-reaction problem. In the contrary to the standard FEM the requirement of the conforming properties is omitted. The discretization is carried out with respect to space variables, whereas time remains continuous. In the discontinuous Galerkin discretization, the nonsymmetric stabilization of diffusion terms combined with interior and boundary penalty is applied. In the evaluation of fluxes the idea of upwinding is used. This allows to obtain an optimal error estimate, also verified by numerical experiments. [ABSTRACT FROM AUTHOR]

Hp-adaptive finite element methods require error estimates of the solution at the current order and one order higher. Hierarchical-based estimation strategies have proved effective in computing errors at the current order for nonlinear parabolic equations. Recently a new approach, interpolation error-based (IEB) error estimation, for constructing a posteriori error estimates at both orders has been developed for linear reaction-diffusion equations. The main results are: (i) IEB error estimation can be applied to nonlinear reaction-diffusion equations in one space dimension; (ii) the hierarchical estimator is an implicit IEB method and thus, works for reaction-diffusion problems; (iii) a hierarchical extension for computing higher-order error estimates is asymptotically exact. Computational results illustrating the theory and comparing the implicit (hierarchical) strategy with the earlier explicit IEB methods are presented. [ABSTRACT FROM AUTHOR]